The ASCA SIS spectrum of the X-ray binary Cygnus X-3 is shown to contain a narrow recombination continuum of hydrogenic sulfur near 3.5 keV, which indicates that the temperature of the X-ray emission-line region is far below the coronal equilibrium temperature required to support S¹⁵⁺. The presence of this feature immediately confirms that X-ray photoionization produces extreme overionization in the circumsource medium, and that the X-ray emission features are energized by the hard X-ray continuum. Therefore, we conclude that spectral analyses of Cygnus X-3 and, most likely, other accretion-powered X-ray sources must be conducted in the context of plasma emission models that account for population kinetics dominated by recombination. Furthermore, the temperature dependence of the shapes of recombination continua provides a simple diagnostic of the electron temperature. We reanalyze the SIS data using a spectral model developed to account for recombination cascades, and, using these diagnostics, derive constraints on the temperatures in the S¹⁵⁺, Si¹³⁺, and Mg¹¹⁺ ionization zones.